Problem 48 - College Algebra Essentials

Question

In Exercises \(45-54,\) rationalize the denominator. $$\frac{\sqrt{7}}{\sqrt{3}}$$

Step-by-Step Solution

Answer

The rationalized form of the fraction is \(\frac{\sqrt{21}}{3}\).

Step 1: Understand what result we want

We need to rationalize the denominator, which means we need to get rid of the square root in the denominator.

Step 2: Multiply by the appropriate fraction

To do this, we can use the property of roots that \(\sqrt{a}*\sqrt{a}=a\) to get rid of the square root in the denominator. We will multiply the numerator and denominator by \(\sqrt{3}\). Thus our fraction becomes \(\frac{\sqrt{7}*\sqrt{3}}{\sqrt{3}*\sqrt{3}}\).

Step 3: Simplify the equation

When we simplify, we get \(\frac{\sqrt{21}}{3}\), a fraction with a rational denominator.

Step 4: Final check

Check that the denominator is now a rational number. If it is, we have completed our task.

Key Concepts

Radical Expression

When dealing with mathematics, one often comes across expressions containing radicals. A radical expression is any mathematical expression involving a radical sign, which indicates the root of a number. The most common radical is the square root, but there are others like cube roots, fourth roots, and so on. The square root of a number, for example \( \sqrt{x} \), is a value that, when multiplied by itself, gives the original number x.

Radical expressions can sometimes be intimidating, but with understanding of their properties, simplifying them becomes much less daunting. In the context of the exercise, \( \frac{\sqrt{7}}{\sqrt{3}} \) is a radical expression with square roots in both the numerator and the denominator. Our goal is to simplify this expression to a form where the radical sign does not appear in the denominator, as this is generally considered to be a more proper form in mathematics.

Square Root Properties

Square roots are subject to specific rules or properties that allow us to manipulate them algebraically. Square root properties are essential for simplifying radical expressions and other algebraic operations. One of the pivotal properties is that the product of square roots is equal to the square root of the product of the numbers, which can be written as \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).

Another property is that the square root of a squared number is the number itself: \( \sqrt{a^2} = a \), assuming 'a' is a non-negative number. This property is utilized when rationalizing the denominator, such as in the exercise, to eliminate the radical on the bottom of a fraction. By understanding and applying these square root properties correctly, one simplifies the expression \( \frac{\sqrt{7}}{\sqrt{3}} \) to \( \frac{\sqrt{21}}{3} \) without the radical in the denominator.

Simplify Fractions

In mathematics, simplification is pivotal for understanding and interpreting numbers or expressions efficiently. To simplify fractions, one must reduce the fraction to its simplest form where the numerator and denominator are as small as possible and have no common factors other than 1. Simplifying fractions makes them easier to work with and to compare.

Fractions involving radicals can also be simplified by rationalizing the denominator. We ensure that the denominator is a rational number (a number that can be expressed as the quotient of two integers). One common method of rationalizing the denominator, particularly when dealing with square roots, is to multiply both the numerator and the denominator by the radical found in the denominator. This multiplication removes the radical in the denominator through the square root property \( \sqrt{a} \times \sqrt{a} = a \), leaving us with a simplified fraction that maintains the same value as the original expression.